Paper · Apr 10, 2020

A Zoo of Translating Solitons on a Parallel Light-like Direction in Minkowski 3-Space

Erdem Kocakuşaklı, Miguel Ortega

math.DGWider flowsarXiv:2004.04974v2
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Abstract

We deal with solitons of the mean curvature flow. The definition of translating solitons on a light-like direction in Minkowski 3-space is introduced. Firstly, we classify those which are graphical, translation surfaces, obtaining space-like and time-like, entire and not entire, complete and incomplete examples. Among them, all our time-like examples are incomplete. The second family consists of those which are invariant by a 1-dimensional subgroup of parabolic motions, i.e, with light-like axis. The classification result implies that all examples of this second family have singularities.

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Submitted
Apr 10, 2020
Last revised
Jul 2, 2021 (v2)
Subjects
math.DG
DOI
10.1016/j.difgeo.2021.101796
Mean curvature flowSelf-shrinkers & translatorsSingularitiesSurfaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Quantitative index bounds for translators via topology (Debora Impera and Michele Rimoldi, 2018)8Impera 2018 Ruled surfaces of generalized self-similar solutions of the mean curvature flow (Rafael López, 2020)4López 2020 Translating Solitons in a Lorentzian Setting, Submersions and Cohomogeneity One Actions (Marie-Amelie Lawn and Miguel Ortega, 2020)1Lawn 2020 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)2Zhu 2020 On the spectrum and index of expanding and translating solitons of the mean curvature flow in ℝ³ (Hilário Alencar and Gregório Silva Neto, 2022)6Alencar 2022 New Examples of Translating Solitons in Generalised Robertson-Walker Geometries (Diego Artacho, Marie-Amélie Lawn and Miguel Ortega, 2022)3Artacho 2022 Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe (Miguel Ortega and Buse Yalçın, 2024)7Ortega 2024 Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in ℍ³ as Translation Surfaces (Tarcios Andrey Ferreira and João Paulo dos Santos, 2025)5Ferreira 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Translating Solitons in a Lorentzian Setting, Submersions and Cohomogeneity One Actions

    Marie-Amelie Lawn and Miguel OrtegaWider flows

    Topics Self-shrinkers & translators · Same author Miguel Ortega · Shared terms “translating solitons”, “Minkowski”

  2. symmetry of the translating solitons of the mean curvature flow in

    Jingze ZhuWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “translating solitons”

  3. New Examples of Translating Solitons in Generalised Robertson-Walker Geometries

    Diego Artacho, Marie-Amélie Lawn and Miguel OrtegaWider flows

    Topics Self-shrinkers & translators · Same author Miguel Ortega · Shared terms “translating solitons”, “translations”

  4. Ruled surfaces of generalized self-similar solutions of the mean curvature flow

    Rafael LópezWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “translation surfaces”

  5. Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in as Translation Surfaces

    Tarcios Andrey Ferreira and João Paulo dos SantosWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “translation surfaces”

  6. On the spectrum and index of expanding and translating solitons of the mean curvature flow in

    Hilário Alencar and Gregório Silva NetoWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “translating solitons”

  7. Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe

    Miguel Ortega and Buse YalçınWider flows

    Topics Self-shrinkers & translators · Same author Miguel Ortega

  8. Quantitative index bounds for translators via topology

    Debora Impera and Michele RimoldiWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “direction”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.